A364758 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)).

1, 1, 3, 14, 76, 450, 2818, 18352, 123028, 843345, 5884227, 41650479, 298352365, 2158751879, 15754446893, 115830820439, 857147952469, 6379136387303, 47715901304501, 358529599468636, 2704884469806606, 20481615947325089, 155605509972859999, 1185779099027494848

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.

From Seiichi Manyama, Dec 11 2024: (Start)

G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 + x*A(x))).

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

Prog

(PARI) a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);
(PARI) a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 11 2024

Crossrefs

Cf. A090192, A106228, A364759, A378919.

Cf. A001764, A219537, A364865, A365224.

Cf. A300048, A364747, A378889.

Sequence in context: A223026 A371434 A364477 * A353253 A198702 A198621

Adjacent sequences: A364755 A364756 A364757 * A364759 A364760 A364761

Keyword

nonn

Author

Seiichi Manyama, Aug 05 2023

Status

approved